Problem: Given that $\binom{23}{3}=1771$, $\binom{23}{4}=8855$, and $\binom{23}{5}=33649$, find $\binom{25}{5}$.
We can use Pascal's identity $\binom{n-1}{k-1}+\binom{n-1}{k}=\binom{n}{k}$ to find $\binom{24}{4}$ and $\binom{24}{5}$.

$$\binom{24}{4}=\binom{23}{3}+\binom{23}{4}=1771+8855=10626$$ $$\binom{24}{5}=\binom{23}{4}+\binom{23}{5}=8855+33649=42504$$

Now that we have $\binom{24}{4}$ and $\binom{24}{5}$, we can use Pascal's identity again to find $\binom{25}{5}$.

$$\binom{25}{5}=\binom{24}{4}+\binom{24}{5}=10626+42504=\boxed{53130}$$